×

On half-space problems for the weakly non-linear discrete Boltzmann equation. (English) Zbl 1193.82032

Summary: Existence of solutions of weakly non-linear half-space problems for the general discrete velocity (with arbitrarily finite number of velocities) model of the Boltzmann equation are studied. The solutions are assumed to tend to an assigned Maxwellian at infinity, and the data for the outgoing particles at the boundary are assigned, possibly linearly depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at infinity is different of -1, 0 and 1) implicit conditions are found. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. Applications to axially symmetric models are studied in more detail.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI